Analytical solutions of incomplete elliptic integrals motivated by the work of Ramanujan

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DOI:

https://doi.org/10.26637/mjm204/005

Abstract

In this paper, we obtain exact solutions of some unsolved incomplete elliptic integrals of first, second and third kinds, given in Entry 7 of Chapter XVII of second notebook of Srinivasa Ramanujan. Furthermore, we generalize these elliptic integrals in the forms of multiple series identities involving bounded multiple sequences.

Keywords:

Gaussian Hypergeometric Function, Incomplete Elliptic Integrals, Multiple Series Identities, Srivastava-Daoust double Hypergeometric Function, Kampé de Fériet double Hypergeometric Function, Sri- vastava’s Triple Hypergeometric Function

Mathematics Subject Classification:

33B10, 33C05, 33C65
  • M. I. Qureshi Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India. https://orcid.org/0000-0002-5093-1925
  • C. W. Mohd Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India.
  • M. P. Chaudhary Former Researcher, Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India. https://orcid.org/0000-0002-7329-2732
  • K. A. Quraishi Former Researcher, Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India.
  • I. H. Khan Former Researcher, Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India.
  • Pages: 373-391
  • Date Published: 01-10-2014
  • Vol. 2 No. 04 (2014): Malaya Journal of Matematik (MJM)

Abramowitz, M. and Stegun, I. A.; Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables, Appl. Math. Ser. 55. U. S. Govt. Printing Office, National Bureau of Standards, Washington D. C., 1964.; Reprinted by Dover Publications, Inc., New York, 1965.

Agarwal, R. P.; Resonance of Ramanujan's Mathematics, Vol. I, New Age International(P) Ltd., New Delhi, 1996.

Agarwal, R. P.; Resonance of Ramanujan's Mathematics, Vol. II, New Age International(P) Ltd., New Delhi, 1996.

Agarwal, R. P.; Resonance of Ramanujan's Mathematics, Vol. III, New Age International(P) Ltd., New Delhi, 1999.

Andrews, G. E. and Berndt, B. C.; Ramanujan's Lost Notebook, Part I, Springer-Verlag, New York, 2005. DOI: https://doi.org/10.1007/0-387-28124-X

Berndt, B. C.; Ramanujan's Notebooks, Part I, Springer-Verlag, New York, 1985. DOI: https://doi.org/10.1007/978-1-4612-1088-7

Berndt, B. C.; Ramanujan's Notebooks, Part II, Springer-Verlag, New York 1989. DOI: https://doi.org/10.1007/978-1-4612-4530-8

Berndt, B. C.; Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991. DOI: https://doi.org/10.1007/978-1-4612-0965-2

Berndt, B. C.; Ramanujan's Notebooks, Part IV, Springer-Verlag, New York 1994. DOI: https://doi.org/10.1007/978-1-4612-0879-2

Berndt, B. C.; Ramanujan's Notebooks, Part V, Springer-Verlag, New York 1998, DOI: https://doi.org/10.1007/978-1-4612-1624-7

Byrd, P. F. and Friedman, M. D.; Handbook of Elliptic Integrals for Engineers and Scientists, Springer -Verlag, Heidelberg, 1971. DOI: https://doi.org/10.1007/978-3-642-65138-0

Carlson, B. C.; Some Series and Bounds for Incomplete Elliptic Integral, J. Math. Phys., 40(1961), 125-134. DOI: https://doi.org/10.1002/sapm1961401125

Carlson, B. C.; Normal Elliptic Integrals of the First and Second Kinds, Duke Math. J., 31(1964), 405-419. DOI: https://doi.org/10.1215/S0012-7094-64-03139-4

Chaudhary, M. P.; On Certain Aspects of Generalized Special Functions and Integral Operators, Doctoral thesis approved for Ph.D. degree, J.M.I.(A Central University), New Delhi, India, 2011.

Denis, R. Y., Singh, S. N. and Singh, S. P.; On Certain Elliptic Integrals of Ramanujan, J. Indian Math. Soc., 37(3-4) (2006), 113-119.

Dutka, J.; Two Results of Ramanujan, Siam J. Math. Anal., 12(1981), 471-476. DOI: https://doi.org/10.1137/0512040

Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.; Higher Transcendental Functions, Vol. II(Bateman Manuscript Project), McGraw-Hill, Book Co. Inc., New York, Toronto and London, 1953.

Gradshteyn, I. S. and Ryzhik, I. M.;Table of Integrals, Series and Products, Fourth Edition, 1965, Corrected and Enlarged Edition by A. Jeffrey 1980, 5 ${ }^{text {th }}$ Ed. by A. Jeffrey, Academic Press, New York, 1994.

Hansen, E.R.; A Table of Series and Product, Prentice Hall, Engle-Wood Cliffs, New Jersey, 1975.

Hardy, G. H., Aiyer, P. V. Seshu and Wilson, B. M.; Collected Papers of Srinivasa Ramanujan, First Published by Cambridge University Press, Cambridge, 1927; Reprinted by Chelsea, New York, 1962; Reprinted by the American Mathematical Society, Providence, Rhode Island, 2000.

Jahnke, E. and Emde, F.; Tables of Functions with Formulas and Curves, Fourth Edition, Dover Publications, Inc., New York, 1945.

Jahnke, E., Emde, F. and Lösch, F.; Tables of Higher Functions, Sixth Edition, McGraw-Hill, New York, 1960.

Jeffrey, A.; Handbook of Mathematical Formulas and Integrals, Second Edition, Academic Press, San Diego, San Francisco, New York, Boston, London, Sydney and Tokyo, 2000.

Magnus, W., Oberhettinger, F. and Soni, R. P.; Formulas and Theorems for the Special Function of Mathematical Physics, Springer-Verlag, New York, 1966. DOI: https://doi.org/10.1007/978-3-662-11761-3

Montaldi, E. and Zucchelli, G.; Some Formulas of Ramanujan Revisited, SIAM J. Math. Anal., 23(1992), $562-569$ DOI: https://doi.org/10.1137/0523028

Pathan, M. A.; Some Formulas of Ramanujan on Hypergeometric Series, The Math. Student, 72(2003), 31-44.

Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I.; Integrals and Series, Vol 3., More Special Functions, Nauka, Moscow, 1986; Translated from the Russian by G. G. Gould; Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, 1990.

Raghavan, S. and Rangachari, S. S.; On Ramanujan's Elliptic Integrals and Modular Identities in Number Theory and Related Topics, Oxford University Press, Bombay, 1989, 119-149.

Ramanujan, S.; Notebooks of Srinivasa Ramanujan, Vol. I, Tata Institute of Fundamental Research, Bombay, 1957; Reprinted by Narosa Publishing House, New Delhi, 1984.

Ramanujan, S.; Notebooks of Srinivasa Ramanujan, Vol. II, Tata Institute of Fundamental Research, Bombay, 1957; Reprinted by Narosa Publishing House, New Delhi, 1984. DOI: https://doi.org/10.1007/978-3-662-30224-8

Ramanujan, S.; The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.

Rao, K. Srinivasa; Srinivasa Ramanujan: A Mathematical Genius, East West Books (Madras) Pvt. Ltd., Chennai, Bangalore, Hyderabad, 1998.

Srivastava, H. M. and Karlsson, Per W.; Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Ltd., Chichester, Brisbane, U. K.) John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.

Srivastava, H. M. and Manocha, H. L.; A Treatise on Generating Functions, Halsted Press (Ellis Horwood Ltd., Chichester, Brisbane, U. K.) John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

Venkatchala, B. J., Vinay, V. and Yogananda, C. S.; Ramanujan's Papers, Prism Books Pvt. Ltd., Bangalore, Mumbai, India, 2000.

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Published

01-10-2014

How to Cite

M. I. Qureshi, C. W. Mohd, M. P. Chaudhary, K. A. Quraishi, and I. H. Khan. “Analytical Solutions of Incomplete Elliptic Integrals Motivated by the Work of Ramanujan”. Malaya Journal of Matematik, vol. 2, no. 04, Oct. 2014, pp. 373-91, doi:10.26637/mjm204/005.